Protocols for Rydberg Entangling Gates Featuring Robustness against Quasistatic Errors

We introduce a novel family of protocols for entangling gates for neutral atom qubits based on the Rydberg blockade mechanism. These protocols realize controlled-phase gates through a series of global laser pulses that are on resonance with the Rydberg excitation frequency. We analyze these protocols with respect to their robustness against calibration errors of the Rabi frequency or shot-to-shot laser intensity fluctuations, and show that they display robustness in various fidelity measures. In addition, we discuss adaptations of these protocols in order to make them robust to atomic-motion-induced Doppler shifts as well.

Popular Summary

Arrays of neutral atoms trapped by optical tweezers have emerged as a promising platform for implementing quantum information processing protocols. These arrays offer unique features, such as a high degree of programmability, high-fidelity manipulation of single qubits via optical control, as well as the realization of entangling gates via excitation to Rydberg states. One of the main challenges in developing this platform for quantum computing applications is increasing the fidelity of multiqubit gates. In this paper, we introduce novel, robust protocols for realizing such gates. In particular, these protocols are robust against certain coherent errors arising from fluctuations in laser intensity or finite temperatures of the atoms in the traps.

The protocols introduced in this paper consist of a sequence of laser pulses with several attractive features: The atoms can be globally addressed by the laser, avoiding challenges associated with local control, and the required laser frequency can be chosen to be exactly resonant with the transition to the Rydberg level, simplifying calibration. Most importantly, the protocols result in gate fidelities that are insensitive to calibration errors and low-frequency fluctuations in laser intensity, as well as errors due to Doppler shifts from thermal motion of the atoms.

Figure 1

Bloch sphere trajectories followed by the effective two-level system $|0,1⟩\leftrightarrow |0,r⟩$ and, equivalently, $|1,0⟩\leftrightarrow |r,0⟩$ (left) and of the effective two-level system $|1,1⟩\leftrightarrow |W⟩$ (right) during sequence (2). States $|0,1⟩$ and $|1,0⟩$ are respectively coupled to $|0,r⟩$ and $|r,0⟩$ with a coupling strength $\mathrm{\Omega }$, while $|1,1⟩$ is coupled to $|W⟩=(|1,r⟩+|r,1⟩)/\sqrt{2}$ with a coupling strength $\sqrt{2}\mathrm{\Omega }$. Both trajectories are closed, with an enclosed area of $2\pi$, such that the three computational basis states $|1,0⟩$, $|0,1⟩$, and $|1,1⟩$ pick up a minus sign. Since state $|0,0⟩$ does not evolve, this realizes a cz gate.

Figure 2

Bloch sphere trajectories for the controlled-$\pi /2$ gate sequence. The phase jump between the third and fourth pulses ensures that the area enclosed in the trajectories now represents one quarter of the sphere, so at the end of the sequence, states $|0,1⟩$, $|1,0⟩$, and $|1,1⟩$ have acquired a phase of $-\pi /2$. Again, $|0,0⟩$ does not evolve, resulting in the desired gate.

Figure 3

Left: pulse sequences corresponding to the cz gate protocols introduced here. Each global pulse is represented by one box, with the corresponding pulse area ${\alpha }_1$ or ${\alpha }_2$. The values of (${\alpha }_1,{\alpha }_2$) can be chosen from two options: $(\pi /(2\sqrt{2}),\pi )$, as in sequence (2), or $(\pi /2,\pi /\sqrt{2})$, as in sequence (3). The laser phase for each pulse is represented by the shape and color of the corresponding box’s border. A filled box means that the detuning error’s sign is flipped for this pulse. The protocol duration is proportional to the number of boxes, and is indicated at the bottom of the figure. Right: values of the leakage (${\chi }_{\mathcal{P}}$) and conditional (${\chi }_{\mathcal{C}}$) susceptibilities for the error parameters associated with the intensity or Doppler error for all the protocols introduced above as well as the protocols of Refs. [5, 9]. The values given here correspond to a cz gate with the choice $({\alpha }_1,{\alpha }_2)=(\pi /(2\sqrt{2}),\pi )$; for an arbitrary phase and/or with $({\alpha }_1,{\alpha }_2)=(\pi /2,\pi /\sqrt{2})$, the specific values can be different but the robustness properties are the same (except for one case, denoted by a star in the table, where leakage robustness is only achieved in the cz case).

Figure 4

Bloch sphere representation of the three-qubit ccz gate sequence. The coupled superposition states $|W_2⟩$ and $|W_3⟩$ are respectively defined as $|W_2⟩=(|0,1,r⟩+|0,r,1⟩)/\sqrt{2}$ and $|W_3⟩=(|1,1,r⟩+|1,r,1⟩+|r,1,1⟩)/\sqrt{3}$; the associated coupling strengths are respectively $\sqrt{2}\mathrm{\Omega }$ and $\sqrt{3}\mathrm{\Omega }$. For all three Bloch spheres, the solid angle enclosed in the trajectory represents half the sphere, so all computational basis states except $|0,0,0⟩$ acquire a phase of $\pi$ during this sequence.

Figure 5

(a) Dependence of the gate fidelity on the ratio of blockade strength $V$ and Rabi frequency $\mathrm{\Omega }$ for the naive implementation of protocol I (gray line) or for the adapted sequences (black dots): the adapted sequences all have $\mathcal{F}=1$. Inset: value of the laser detuning for the adapted sequences, depending on $\mathrm{\Omega }/V$. (b) Susceptibilities to intensity errors as a function of the blockade strength for the adapted versions of protocols I (${\chi }_{\mathcal{C}}$, blue) and II ($\chi$, red). Red and blue solid lines serve as a guide to highlight the inverse square dependence. Red and blue dashed lines are respectively the full and conditional susceptibilities of the protocol of Ref.  [9] for perfect blockade (see Fig. 3).

Figure 6

cz pulse sequences for protocols I.a ($T_{\mathrm{I}}=10.73/\mathrm{\Omega }$) and II.b ($T_{\mathrm{II}}=21.45/\mathrm{\Omega }$), and susceptibilities of these sequences against intensity and Doppler errors.